[Cheng-Wei Chiang, E.S., 1707.06765 (PLB)]

### ξ dependence of V

eff### propagates to GW spectrum significantly!

### e.g., scale-inv. U(1)

B-L### model

no resum ⇠ = 0 ⇠ = 1 ⇠ = 5

↵ 2.27 1.44 0.99 0.48

˜ 89.4 97.5 105.4 135.0

phase transition strength and the GW spectrum are to the gauge-fixing parameter ⇠ using an explicit model.

After the thermal resummation, one cannot completely gauge away the kinetic energy of the gauge field. However, since such an energy is gauge-independent, we will neglect it in the following discussions for simplicity. Furthermore, the critical bubble for the first-order phase transition in the early Universe is assumed to be spherically symmetric, with the energy functional given by

S_{3} = 4⇡

Z _{1}

0

dr r^{2}

1 2

✓d _{S}
dr

◆2

+ V_{e↵}( _{S}; T ) , (15)

where _{S}(r) = p

2hS(r)i. The equation of motion for ^{S} is then
d^{2} _{S}

dr^{2} + 2
r

d _{S}
dr

@V_{e↵}

@ _{S} = 0 , (16)

with the boundary conditions: lim_{r}_{!1} _{S}(r) = 0 and d _{S}(r)/dr|^{r=0} = 0. We can solve
Eq. (16) by use of a relaxation method (see, e.g., Ref. [30] for details).

Let T_{⇤} be the temperature at which the GWs are produced from the cosmological phase
transition. Without significant reheating, this temperature can be approximated by the
bubble nucleation temperature, T_{N}, to be defined below. For the phase transition to develop,
at least one bubble must nucleate within the Hubble volume. We thus define T_{N} through
the condition

N(T_{N}) = H^{4}(T_{N}) , (17)

where H(T ) = 1.66p

g_{⇤}(T )T^{2}/m_{Pl} with g_{⇤}(T ) being the relativistic degrees of freedom at
T and m_{Pl} = 1.22 ⇥ 10^{19} GeV, while _{N}(T ) is the bubble nucleation rate per unit time per
unit volume approximately given by [31]

N(T ) ' T^{4}

✓S_{3}(T )
2⇡T

◆3/2

e ^{S}^{3}^{(T )/T} . (18)

From Eqs. (17) and (18), one obtains S_{3}(T_{N})/T_{N} ' 140 150.

A model-independent analysis of the GWs has been done in Ref. [6] using two parameters:

↵ ⌘ ✏(T_{⇤})

⇢_{rad}(T_{⇤}) and ⌘ H⇤T_{⇤} d
dT

✓S_{3}(T )
T

◆

T =T_{⇤}

, (19)

where

✏(T ) = V_{e↵} T @ V_{e↵}

@T and ⇢_{rad}(T ) = ⇡^{2}

30g_{⇤}(T )T^{4}, (20)
7

phase transition strength and the GW spectrum are to the gauge-fixing parameter ⇠ using an explicit model.

After the thermal resummation, one cannot completely gauge away the kinetic energy of the gauge field. However, since such an energy is gauge-independent, we will neglect it in the following discussions for simplicity. Furthermore, the critical bubble for the first-order phase transition in the early Universe is assumed to be spherically symmetric, with the energy functional given by

S3 = 4⇡

Z _{1}

0

dr r^{2}

1 2

✓d _{S}
dr

◆2

+ Ve↵( S; T ) , (15) where S(r) = p

2hS(r)i. The equation of motion for ^{S} is then
d^{2} _{S}

dr^{2} + 2
r

d _{S}
dr

@V_{e↵}

@ _{S} = 0 , (16)

with the boundary conditions: lim_{r}_{!1 S}(r) = 0 and d _{S}(r)/dr|^{r=0} = 0. We can solve
Eq. (16) by use of a relaxation method (see, e.g., Ref. [30] for details).

Let T_{⇤} be the temperature at which the GWs are produced from the cosmological phase
transition. Without significant reheating, this temperature can be approximated by the
bubble nucleation temperature, TN, to be defined below. For the phase transition to develop,
at least one bubble must nucleate within the Hubble volume. We thus define T_{N} through
the condition

N(T_{N}) = H^{4}(T_{N}) , (17)

where H(T ) = 1.66p

g_{⇤}(T )T^{2}/mPl with g_{⇤}(T ) being the relativistic degrees of freedom at
T and m_{Pl} = 1.22 ⇥ 10^{19} GeV, while _{N}(T ) is the bubble nucleation rate per unit time per
unit volume approximately given by [31]

N(T ) ' T^{4}

✓S3(T ) 2⇡T

◆3/2

e ^{S}^{3}^{(T )/T} . (18)

From Eqs. (17) and (18), one obtains S3(TN)/TN ' 140 150.

A model-independent analysis of the GWs has been done in Ref. [6] using two parameters:

↵ ⌘ ✏(T_{⇤})

⇢rad(T_{⇤}) and ⌘ H⇤T_{⇤} d
dT

✓S_{3}(T )
T

◆

T =T_{⇤}

, (19)

where

✏(T ) = V_{e↵} T @ Ve↵

@T and ⇢_{rad}(T ) = ⇡^{2}

30g_{⇤}(T )T^{4}, (20)
7

10^{-18}
10^{-16}
10^{-14}
10^{-12}
10^{-10}
10^{-8}
10^{-6}

10^{-6} 10^{-5} 10^{-4} 10^{-3} 10^{-2} 10^{-1} 10^{0} 10^{1} 10^{2}

f [Hz]

ΩGWh2

no resum ξ = 0 ξ = 1 ξ = 5

**Impact of ξ on gravitational wave**

[Cheng-Wei Chiang, E.S., 1707.06765 (PLB)]

### ξ dependence of V

eff### propagates to GW spectrum significantly!

### e.g., scale-inv. U(1)

B-L### model

### ξ =0

no resum ⇠ = 0 ⇠ = 1 ⇠ = 5

↵ 2.27 1.44 0.99 0.48

˜ 89.4 97.5 105.4 135.0

phase transition strength and the GW spectrum are to the gauge-fixing parameter ⇠ using an explicit model.

After the thermal resummation, one cannot completely gauge away the kinetic energy of the gauge field. However, since such an energy is gauge-independent, we will neglect it in the following discussions for simplicity. Furthermore, the critical bubble for the first-order phase transition in the early Universe is assumed to be spherically symmetric, with the energy functional given by

S_{3} = 4⇡

Z _{1}

0

dr r^{2}

1 2

✓d _{S}
dr

◆2

+ V_{e↵}( _{S}; T ) , (15)

where _{S}(r) = p

2hS(r)i. The equation of motion for ^{S} is then
d^{2} _{S}

dr^{2} + 2
r

d _{S}
dr

@V_{e↵}

@ _{S} = 0 , (16)

with the boundary conditions: lim_{r}_{!1} _{S}(r) = 0 and d _{S}(r)/dr|^{r=0} = 0. We can solve
Eq. (16) by use of a relaxation method (see, e.g., Ref. [30] for details).

Let T_{⇤} be the temperature at which the GWs are produced from the cosmological phase
transition. Without significant reheating, this temperature can be approximated by the
bubble nucleation temperature, T_{N}, to be defined below. For the phase transition to develop,
at least one bubble must nucleate within the Hubble volume. We thus define T_{N} through
the condition

N(T_{N}) = H^{4}(T_{N}) , (17)

where H(T ) = 1.66p

g_{⇤}(T )T^{2}/m_{Pl} with g_{⇤}(T ) being the relativistic degrees of freedom at
T and m_{Pl} = 1.22 ⇥ 10^{19} GeV, while _{N}(T ) is the bubble nucleation rate per unit time per
unit volume approximately given by [31]

N(T ) ' T^{4}

✓S_{3}(T )
2⇡T

◆3/2

e ^{S}^{3}^{(T )/T} . (18)

From Eqs. (17) and (18), one obtains S_{3}(T_{N})/T_{N} ' 140 150.

A model-independent analysis of the GWs has been done in Ref. [6] using two parameters:

↵ ⌘ ✏(T_{⇤})

⇢_{rad}(T_{⇤}) and ⌘ H⇤T_{⇤} d
dT

✓S_{3}(T )
T

◆

T =T_{⇤}

, (19)

where

✏(T ) = V_{e↵} T @ V_{e↵}

@T and ⇢_{rad}(T ) = ⇡^{2}

30g_{⇤}(T )T^{4}, (20)
7

S3 = 4⇡

Z _{1}

0

dr r^{2}

1 2

✓d _{S}
dr

◆2

+ Ve↵( S; T ) , (15) where S(r) = p

2hS(r)i. The equation of motion for ^{S} is then
d^{2} _{S}

dr^{2} + 2
r

d _{S}
dr

@V_{e↵}

@ _{S} = 0 , (16)

with the boundary conditions: lim_{r}_{!1 S}(r) = 0 and d _{S}(r)/dr|^{r=0} = 0. We can solve
Eq. (16) by use of a relaxation method (see, e.g., Ref. [30] for details).

Let T_{⇤} be the temperature at which the GWs are produced from the cosmological phase
transition. Without significant reheating, this temperature can be approximated by the
bubble nucleation temperature, TN, to be defined below. For the phase transition to develop,
at least one bubble must nucleate within the Hubble volume. We thus define T_{N} through
the condition

N(T_{N}) = H^{4}(T_{N}) , (17)

where H(T ) = 1.66p

g_{⇤}(T )T^{2}/mPl with g_{⇤}(T ) being the relativistic degrees of freedom at
T and m_{Pl} = 1.22 ⇥ 10^{19} GeV, while _{N}(T ) is the bubble nucleation rate per unit time per
unit volume approximately given by [31]

N(T ) ' T^{4}

✓S3(T ) 2⇡T

◆3/2

e ^{S}^{3}^{(T )/T} . (18)

From Eqs. (17) and (18), one obtains S3(TN)/TN ' 140 150.

A model-independent analysis of the GWs has been done in Ref. [6] using two parameters:

↵ ⌘ ✏(T_{⇤})

⇢rad(T_{⇤}) and ⌘ H⇤T_{⇤} d
dT

✓S_{3}(T )
T

◆

T =T_{⇤}

, (19)

where

✏(T ) = V_{e↵} T @ Ve↵

@T and ⇢_{rad}(T ) = ⇡^{2}

30g_{⇤}(T )T^{4}, (20)
7

10^{-18}
10^{-16}
10^{-14}
10^{-12}
10^{-10}
10^{-8}
10^{-6}

10^{-6} 10^{-5} 10^{-4} 10^{-3} 10^{-2} 10^{-1} 10^{0} 10^{1} 10^{2}

f [Hz]

ΩGWh2

no resum ξ = 0 ξ = 1 ξ = 5

**Impact of ξ on gravitational wave**

[Cheng-Wei Chiang, E.S., 1707.06765 (PLB)]

### ξ dependence of V

eff### propagates to GW spectrum significantly!

### e.g., scale-inv. U(1)

B-L### model

### ξ =0

no resum ⇠ = 0 ⇠ = 1 ⇠ = 5

↵ 2.27 1.44 0.99 0.48

˜ 89.4 97.5 105.4 135.0

S_{3} = 4⇡

Z _{1}

0

dr r^{2}

1 2

✓d _{S}
dr

◆2

+ V_{e↵}( _{S}; T ) , (15)

where _{S}(r) = p

2hS(r)i. The equation of motion for ^{S} is then
d^{2} _{S}

dr^{2} + 2
r

d _{S}
dr

@V_{e↵}

@ _{S} = 0 , (16)

with the boundary conditions: lim_{r}_{!1} _{S}(r) = 0 and d _{S}(r)/dr|^{r=0} = 0. We can solve
Eq. (16) by use of a relaxation method (see, e.g., Ref. [30] for details).

Let T_{⇤} be the temperature at which the GWs are produced from the cosmological phase
transition. Without significant reheating, this temperature can be approximated by the
bubble nucleation temperature, T_{N}, to be defined below. For the phase transition to develop,
at least one bubble must nucleate within the Hubble volume. We thus define T_{N} through
the condition

N(T_{N}) = H^{4}(T_{N}) , (17)

where H(T ) = 1.66p

g_{⇤}(T )T^{2}/m_{Pl} with g_{⇤}(T ) being the relativistic degrees of freedom at
T and m_{Pl} = 1.22 ⇥ 10^{19} GeV, while _{N}(T ) is the bubble nucleation rate per unit time per
unit volume approximately given by [31]

N(T ) ' T^{4}

✓S_{3}(T )
2⇡T

◆3/2

e ^{S}^{3}^{(T )/T} . (18)

From Eqs. (17) and (18), one obtains S_{3}(T_{N})/T_{N} ' 140 150.

A model-independent analysis of the GWs has been done in Ref. [6] using two parameters:

↵ ⌘ ✏(T_{⇤})

⇢_{rad}(T_{⇤}) and ⌘ H⇤T_{⇤} d
dT

✓S_{3}(T )
T

◆

T =T_{⇤}

, (19)

where

✏(T ) = V_{e↵} T @ V_{e↵}

@T and ⇢_{rad}(T ) = ⇡^{2}

30g_{⇤}(T )T^{4}, (20)
7

S3 = 4⇡

Z _{1}

0

dr r^{2}

1 2

✓d _{S}
dr

◆2

+ Ve↵( S; T ) , (15) where S(r) = p

2hS(r)i. The equation of motion for ^{S} is then
d^{2} _{S}

dr^{2} + 2
r

d _{S}
dr

@V_{e↵}

@ _{S} = 0 , (16)

with the boundary conditions: lim_{r}_{!1 S}(r) = 0 and d _{S}(r)/dr|^{r=0} = 0. We can solve
Eq. (16) by use of a relaxation method (see, e.g., Ref. [30] for details).

Let T_{⇤} be the temperature at which the GWs are produced from the cosmological phase
transition. Without significant reheating, this temperature can be approximated by the
bubble nucleation temperature, TN, to be defined below. For the phase transition to develop,
at least one bubble must nucleate within the Hubble volume. We thus define T_{N} through
the condition

N(T_{N}) = H^{4}(T_{N}) , (17)

where H(T ) = 1.66p

g_{⇤}(T )T^{2}/mPl with g_{⇤}(T ) being the relativistic degrees of freedom at
T and m_{Pl} = 1.22 ⇥ 10^{19} GeV, while _{N}(T ) is the bubble nucleation rate per unit time per
unit volume approximately given by [31]

N(T ) ' T^{4}

✓S3(T ) 2⇡T

◆3/2

e ^{S}^{3}^{(T )/T} . (18)

From Eqs. (17) and (18), one obtains S3(TN)/TN ' 140 150.

A model-independent analysis of the GWs has been done in Ref. [6] using two parameters:

↵ ⌘ ✏(T_{⇤})

⇢rad(T_{⇤}) and ⌘ H⇤T_{⇤} d
dT

✓S_{3}(T )
T

◆

T =T_{⇤}

, (19)

where

✏(T ) = V_{e↵} T @ Ve↵

@T and ⇢_{rad}(T ) = ⇡^{2}

30g_{⇤}(T )T^{4}, (20)
7

10^{-18}
10^{-16}
10^{-14}
10^{-12}
10^{-10}
10^{-8}
10^{-6}

10^{-6} 10^{-5} 10^{-4} 10^{-3} 10^{-2} 10^{-1} 10^{0} 10^{1} 10^{2}

f [Hz]

ΩGWh2

no resum ξ = 0 ξ = 1 ξ = 5

**Impact of ξ on gravitational wave**

[Cheng-Wei Chiang, E.S., 1707.06765 (PLB)]

### ξ dependence of V

eff### propagates to GW spectrum significantly!

### e.g., scale-inv. U(1)

B-L### model

### ξ =0 ξ =5

no resum ⇠ = 0 ⇠ = 1 ⇠ = 5

↵ 2.27 1.44 0.99 0.48

˜ 89.4 97.5 105.4 135.0

S_{3} = 4⇡

Z _{1}

0

dr r^{2}

1 2

✓d _{S}
dr

◆2

+ V_{e↵}( _{S}; T ) , (15)

where _{S}(r) = p

2hS(r)i. The equation of motion for ^{S} is then
d^{2} _{S}

dr^{2} + 2
r

d _{S}
dr

@V_{e↵}

@ _{S} = 0 , (16)

_{r}_{!1} _{S}(r) = 0 and d _{S}(r)/dr|^{r=0} = 0. We can solve
Eq. (16) by use of a relaxation method (see, e.g., Ref. [30] for details).

_{⇤} be the temperature at which the GWs are produced from the cosmological phase
transition. Without significant reheating, this temperature can be approximated by the
bubble nucleation temperature, T_{N}, to be defined below. For the phase transition to develop,
at least one bubble must nucleate within the Hubble volume. We thus define T_{N} through
the condition

N(T_{N}) = H^{4}(T_{N}) , (17)

where H(T ) = 1.66p

_{⇤}(T )T^{2}/m_{Pl} with g_{⇤}(T ) being the relativistic degrees of freedom at
T and m_{Pl} = 1.22 ⇥ 10^{19} GeV, while _{N}(T ) is the bubble nucleation rate per unit time per
unit volume approximately given by [31]

N(T ) ' T^{4}

✓S_{3}(T )
2⇡T

◆3/2

e ^{S}^{3}^{(T )/T} . (18)

From Eqs. (17) and (18), one obtains S_{3}(T_{N})/T_{N} ' 140 150.

A model-independent analysis of the GWs has been done in Ref. [6] using two parameters:

↵ ⌘ ✏(T_{⇤})

⇢_{rad}(T_{⇤}) and ⌘ H⇤T_{⇤} d
dT

✓S_{3}(T )
T

◆

T =T_{⇤}

, (19)

where

✏(T ) = V_{e↵} T @ V_{e↵}

@T and ⇢_{rad}(T ) = ⇡^{2}

30g_{⇤}(T )T^{4}, (20)
7

S3 = 4⇡

Z _{1}

0

dr r^{2}

1 2

✓d _{S}
dr

◆2

+ Ve↵( S; T ) , (15) where S(r) = p

2hS(r)i. The equation of motion for ^{S} is then
d^{2} _{S}

dr^{2} + 2
r

d _{S}
dr

@V_{e↵}

@ _{S} = 0 , (16)

_{r}_{!1 S}(r) = 0 and d _{S}(r)/dr|^{r=0} = 0. We can solve
Eq. (16) by use of a relaxation method (see, e.g., Ref. [30] for details).

_{⇤} be the temperature at which the GWs are produced from the cosmological phase
transition. Without significant reheating, this temperature can be approximated by the
bubble nucleation temperature, TN, to be defined below. For the phase transition to develop,
at least one bubble must nucleate within the Hubble volume. We thus define T_{N} through
the condition

N(T_{N}) = H^{4}(T_{N}) , (17)

where H(T ) = 1.66p

_{⇤}(T )T^{2}/mPl with g_{⇤}(T ) being the relativistic degrees of freedom at
T and m_{Pl} = 1.22 ⇥ 10^{19} GeV, while _{N}(T ) is the bubble nucleation rate per unit time per
unit volume approximately given by [31]

N(T ) ' T^{4}

✓S3(T ) 2⇡T

◆3/2

e ^{S}^{3}^{(T )/T} . (18)

From Eqs. (17) and (18), one obtains S3(TN)/TN ' 140 150.

A model-independent analysis of the GWs has been done in Ref. [6] using two parameters:

↵ ⌘ ✏(T_{⇤})

⇢rad(T_{⇤}) and ⌘ H⇤T_{⇤} d
dT

✓S_{3}(T )
T

◆

T =T_{⇤}

, (19)

where

✏(T ) = V_{e↵} T @ Ve↵

@T and ⇢_{rad}(T ) = ⇡^{2}

30g_{⇤}(T )T^{4}, (20)
7

10^{-18}
10^{-16}
10^{-14}
10^{-12}
10^{-10}
10^{-8}
10^{-6}

10^{-6} 10^{-5} 10^{-4} 10^{-3} 10^{-2} 10^{-1} 10^{0} 10^{1} 10^{2}

f [Hz]

ΩGWh2

no resum ξ = 0 ξ = 1 ξ = 5

**Impact of ξ on gravitational wave**

[Cheng-Wei Chiang, E.S., 1707.06765 (PLB)]

### ξ dependence of V

eff### propagates to GW spectrum significantly!

### e.g., scale-inv. U(1)

B-L### model

### ~ 1 order change!!

### ξ =0 ξ =5

no resum ⇠ = 0 ⇠ = 1 ⇠ = 5

↵ 2.27 1.44 0.99 0.48

˜ 89.4 97.5 105.4 135.0

S_{3} = 4⇡

Z _{1}

0

dr r^{2}

1 2

✓d _{S}
dr

◆2

+ V_{e↵}( _{S}; T ) , (15)

where _{S}(r) = p

2hS(r)i. The equation of motion for ^{S} is then
d^{2} _{S}

dr^{2} + 2
r

d _{S}
dr

@V_{e↵}

@ _{S} = 0 , (16)

_{r}_{!1} _{S}(r) = 0 and d _{S}(r)/dr|^{r=0} = 0. We can solve
Eq. (16) by use of a relaxation method (see, e.g., Ref. [30] for details).

_{⇤} be the temperature at which the GWs are produced from the cosmological phase
transition. Without significant reheating, this temperature can be approximated by the
bubble nucleation temperature, T_{N}, to be defined below. For the phase transition to develop,
at least one bubble must nucleate within the Hubble volume. We thus define T_{N} through
the condition

N(T_{N}) = H^{4}(T_{N}) , (17)

where H(T ) = 1.66p

_{⇤}(T )T^{2}/m_{Pl} with g_{⇤}(T ) being the relativistic degrees of freedom at
T and m_{Pl} = 1.22 ⇥ 10^{19} GeV, while _{N}(T ) is the bubble nucleation rate per unit time per
unit volume approximately given by [31]

N(T ) ' T^{4}

✓S_{3}(T )
2⇡T

◆3/2

e ^{S}^{3}^{(T )/T} . (18)

From Eqs. (17) and (18), one obtains S_{3}(T_{N})/T_{N} ' 140 150.

A model-independent analysis of the GWs has been done in Ref. [6] using two parameters:

↵ ⌘ ✏(T_{⇤})

⇢_{rad}(T_{⇤}) and ⌘ H⇤T_{⇤} d
dT

✓S_{3}(T )
T

◆

T =T_{⇤}

, (19)

where

✏(T ) = V_{e↵} T @ V_{e↵}

@T and ⇢_{rad}(T ) = ⇡^{2}

30g_{⇤}(T )T^{4}, (20)
7

S3 = 4⇡

Z _{1}

0

dr r^{2}

1 2

✓d _{S}
dr

◆2

+ Ve↵( S; T ) , (15) where S(r) = p

2hS(r)i. The equation of motion for ^{S} is then
d^{2} _{S}

dr^{2} + 2
r

d _{S}
dr

@V_{e↵}

@ _{S} = 0 , (16)

_{r}_{!1 S}(r) = 0 and d _{S}(r)/dr|^{r=0} = 0. We can solve
Eq. (16) by use of a relaxation method (see, e.g., Ref. [30] for details).

_{⇤} be the temperature at which the GWs are produced from the cosmological phase
transition. Without significant reheating, this temperature can be approximated by the
bubble nucleation temperature, TN, to be defined below. For the phase transition to develop,
at least one bubble must nucleate within the Hubble volume. We thus define T_{N} through
the condition

N(T_{N}) = H^{4}(T_{N}) , (17)

where H(T ) = 1.66p

_{⇤}(T )T^{2}/mPl with g_{⇤}(T ) being the relativistic degrees of freedom at
T and m_{Pl} = 1.22 ⇥ 10^{19} GeV, while _{N}(T ) is the bubble nucleation rate per unit time per
unit volume approximately given by [31]

N(T ) ' T^{4}

✓S3(T ) 2⇡T

◆3/2

e ^{S}^{3}^{(T )/T} . (18)

From Eqs. (17) and (18), one obtains S3(TN)/TN ' 140 150.

A model-independent analysis of the GWs has been done in Ref. [6] using two parameters:

↵ ⌘ ✏(T_{⇤})

⇢rad(T_{⇤}) and ⌘ H⇤T_{⇤} d
dT

✓S_{3}(T )
T

◆

T =T_{⇤}

, (19)

where

✏(T ) = V_{e↵} T @ Ve↵

@T and ⇢_{rad}(T ) = ⇡^{2}

30g_{⇤}(T )T^{4}, (20)
7